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Particle Filter Track BeforeDetect AlgorithmsTheory and Applications
Y. Boers and J.N. Driessen
JRS-PE-FAA
THALES NEDERLAND
Hengelo
The Netherlands
Email:
{yvo.boers,hans.driessen}@nl.thalesgroup.com
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Outline
• Introduction
• Filtering
• Detection
• Examples
• Overview
• Conclusions
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Introduction
Classical vs. TBD
Detect Cluster Extract Track- - - - -
’Raw Data’ Tracks
TBD
TBD integrates the information over time.
Detection is based on power/energy that has
been integrated over time (multiple scans).
Classical tracking : single scan based detec-
tion.
* TBD provides higher probability of detec-
tion (Pd) at the same level of probability of
false alarm (Pfa)
* TBD circumvents the data association prob-
lem.
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Introduction
Twofold problem
The TBD problem is twofold:
1. Filtering
2. Detection
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Filtering
The System
sk+1 = f(tk, sk, dk, wk), k ∈ N
Prob{dk+1 = j | dk = i} = [Π(tk)]ij
zk = h(tk, sk, dk, vk), k ∈ N
Filtering Problem: Determine p(sk, dk|Zk)
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Filtering
Basic idea of the particle filter
” Describe the a posteriori pdf p(sk, dk|Zk) by
a cloud of N particles that propagates in time
such that the cloud approximately equals an
N-sample drawn from p(sk, dk|Zk) ”
NOTE:
This is more than just a (point) estimate !!!!
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FilteringKalman vs. PF representation
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FilteringUsing a (proper) particle filter on the sys-
tem:
The following holds
N∑
i=1
1
Nδ(s− si
k)a.s.−→ p(sk|Zk)
i.e. almost sure convergence...
Popular (point) estimators obtained from par-
ticle cloud:
sMVk =
∫
Rnskp(sk | Zk)dsk ≈
N∑
i=1
1
Nsik
sMAPk = arg max
sk∈Rnp(sk | Zk) ≈ si∗
k
where i∗ = argmaxi qik
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Detection
Deciding upon presence of target:
Hypothesis testing:
Given two hypotheses
• H0 : no signal present
z(j) = v(j), j = 0, . . . , k
• H1 : signal present
z(j) = h(s(j), v(j)), j = 0, . . . , k
where s(k) evolves according to dynamical
system
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Detection
Using particle filter output for detection
Every optimal detector can be expressed in
terms of a Likelihood Ratio Test:
L(Z(k, l)) ≶ τ
THEOREM:
L(Z(k, l)) =p(z(k − l + 1), . . . , z(k) | H1)
p(z(k − l + 1), . . . , z(k) | H0)≈
≈∏k
j=k−l+1(∑N
i=1 qi(j))
N l ∏kj=k−l+1 pv(z(j))
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Detection
Using particle filter output for detection
Elements of the Proof:
p(z(l), . . . , z(m) | H0) =m∏
j=l
pv(zj)
p(z(l), . . . , z(m) | H1) =m∏
j=l
p(z(j) | Z(j − 1))
where
p(z(j) | Z(j − 1)) =∫
Sp(z(j), s | Z(j − 1))ds
=∫
Sp(z(j) | s, Z(j − 1))p(s | Z(j − 1))ds
= Ep(s|Z(j−1))p(z(j) | s, Z(j − 1))
≈ 1
N
N∑
i=1
p(z(j) | si(j)) =1
N
N∑
i=1
qi(j)
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Example - Detection
Linear Gaussian scalar system:
s(k + 1) = s(k) + w(k)
z(k) = s(k) + v(k)
w(k) ∼ N(0,1), v(k) ∼ N(0,1) and s(0) ∼N(0,10)
Data has been generated according to the above
model.
Particle filter solution (200 particles) and the
exact (Kalman) solution have been calcu-
lated.
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Example - DetectionTrue states and estimates
0 5 10 15 20 256
8
10
12
14
16
18
20
22
Ratio exact and p.f. likelihood
0 5 10 15 20 25 30 35 40 45 50
0.95
1
1.05
1.1
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Example - MTT TBD
The Fighter-Missile Example:
Multi target track before detect application
for small to very small closely spaced targets.
Early detection is crucial.
Modelling details in:
Y. Boers, J.N. Driessen, F. Verschure, W.P.M.H. Heemels
and A. Juloski. A Multi Target Track Before Detect Ap-
plication. Workshop on Multi Object Tracking, Madi-
son, WI, June 2003.
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Example - MTT TBD
System:
sk+1 = f(tk, sk, dk) + g(tk, sk, dk)wk
where
f(tk, sk, dk) =
1 0 T 00 1 0 T0 0 1 00 0 0 1
sk
The process noise input model is given by
g(tk, sk, dk) =
12(
13ax,max)T2 0
0 12(
13ay,max)T2
13ax,maxT 0
0 13ay,maxT
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Example- MTT TBDSystem:
The discrete mode dk represents one of three
hypotheses (each have a different measure-
ment equation!!)
• dk = 0: There is no target present.
• dk = 1: The prime target is present.
• dk = 2: There are two targets present.
Markov process:
Π(tk) =
0.90 0.10 0.000.10 0.80 0.100.00 0.10 0.90
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Example - MTT TBD
Simulations
Initially, there is no target present. The first
target (fighter: SNR=13dB) appears after 5
seconds (T = 1s) and moves at a constant
velocity of 200ms−1 towards the sensor.
After 20 seconds, a second target (missile:
SNR 3dB) spawns from the first and accel-
erates to a velocity of 300ms−1 in 3 scans.
1000 particles have been used in a ’plain vanilla
particle filter implementation’
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Example - MTT TBD
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1Estimation of the mode
Time
Pro
bability
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
Time
Pro
bability
True mode
no target present1 target present 2 targets present
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Overview
Related work (co)authored by presenter:
Y. Boers and J.N. Driessen. A Particle Filter Based De-
tection Scheme. IEEE Signal Processing Letters, Oc-
tober, 2003.
Y. Boers, J.N. Driessen, F. Verschure, W.P.M.H. Heemels
and A. Juloski. A Multi Target Track Before Detect Ap-
plication. Workshop on Multi Object Tracking, Madi-
son, WI, June 2003.
Y. Boers and J.N. Driessen. Hybrid State Estimation:
A Target Tracking Application. Automatica, vol. 38,
no.12, 2002.
Y. Boers and J.N. Driessen. An Interacting Multiple
Model Particle Filter. To appear in IEE Proceedings -
Radar, Sonar and Navigation, 2004.
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Overview
Some Other Related work:
D.J. Ballantyne, H. Y. Chan and M.A. Kouritzin. A
novel branching particle method for tracking. SPIE
Aerosense 2000 proceedings, volume 4048, pp. 277-
287, Orlando FL, 24-28 April 2000.
D.J. Salmond and H. Birch, A Particle Filter for Track-
Before-Detect, In Proc. of the American Control Con-
ference, June 25-27, 2001, Arlington, VA.
C. Kreucher et al., Multi Target Tracking Using A Par-
ticle Representation of The Joint Multi Target Density.
Submitted to IEEE Transactions AES / In Proceedings
of SPIE Small Targets Conference, 2003
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Overview
General Particle Filter literature:
As an excellent general book on Particle Fil-
tering with a lot of theory, applications and
references:
A. Doucet, N.J. Gordon and N. de Freitas eds.
Sequential Monte Carlo Methods in Practice,
Springer Verlag, New York, 2001.
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Conclusions
Specific Conclusions
Every optimal detector can be expressed in
terms of PF weights.....Very important result
both from a theoretical and practical point of
view.
A multi target particle filter for closely spaced
targets has been presented for a TBD appli-
cation. The algorithm can be applied in real
time.
Questions/Remarks/Discussions
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